Frontal boundary analysis

Figure 2.2 Classical boundary conditions of common chromatographic problems, (a) Elution of a rectangular pulse, (b) Multiple gradient elution of a rectangular pulse, (c) Staircase frontal analysis, (d) Displacement. Dotted line component 1. Shaded line component 2.

This model has been used by Thomas [83], Goldstein [84], and Wade ei al. [80]. Thomas has derived an analytical solution for a step function boundary condition (i.e., a breakthrough curve or frontal analysis problem) [83]. Goldstein [84] and Wade et al. [80] have derived analytical solutions for pulse boimdary conditions the overloaded elution problem). 

Figure 3.37 Typical experimental chromatograms obtained in the determination of equilibrium isotherms by chromatographic methods, (a) Frontal analysis staircase. FACP on the diffuse rear boundary after the last frontal step, (b) ECP. Data recorded with an HP 1090 (Hewlett-Packard, Palo Alto, CA) liquid chromatograph. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, Anal Chem., 60 (1988) 2630 (Figs. 1 and 2), ( )1988 American Chemical Society.

In the case of a linear isotherm, the solution of the linear ideal model is trivial for the boundary conditions of elution or frontal analysis. This solution is the boundary condition transported along the column at a velocity that is constant. In the case of SMB, the cychc nature of the process makes the solution more complex to derive but also most useful as it predicts with a reasonable precision the concentration profiles and concentration histories. 

Simple Wave Particular boundary conditions for which the solution of the nonideal chromatographic models is mathematically simple. In the case of a breakthrough curve, in frontal analysis or in the injection of a wide rectangular pulse, the concentration of each component varies between two constant values. The solution is said to be a simple wave solution, by reference to the theory of wave propagation which is governed by a similar equation. 

A free boundary model is ui d to describe frontal polymerization. Weakly nonlinear analysis is applied to investigate pulsating instabilities in two dimensions, llie analysis produces a pair of Landau equations, which describe the evolution of the linearly unstable modes. Onset and stability of spinning and standing modes is described. 

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